CK-12-Pre-Calculus Concepts

http://www.ck12.org Chapter 3. Logs and Exponents

unit rate might be higher. Suppose you leave 15% of the algae in the tank and harvest when it reaches 85%. How
much time will that take to yield 70%?

0. 15 = 1 + 19 ·( 01. 74495 )x

x 1 ≈ 4. 10897

0. 85 = 1 + 19 ·( 01. 74495 )x

x 2 ≈ 15. 8914

It takes about 12 days for the batches to yield 70% harvest which is a unit rate of about 6% per day. This is a
significant increase in efficiency. A harvest schedule that maximizes the time where the logistic curve is steepest
creates the fastest overall algae growth.
Concept Problem Revisited
The logistic model is appropriate whenever the total count has an upper limit and the initial growth is exponen-
tial. Examples are the spread of rumors and disease in a limited population and the growth of bacteria or human
population when resources are limited.

Vocabulary

Carrying capacityis the maximum sustainable population that the environmental factors will support. In other
words, it is the population limit.

Guided Practice

1. Given the following logistic model, predict thexvalue that will produce a height of 14 and then predict the height
   whenxis 4.
   f(x) = 1 + 420 ·( 0. 9 )x


2. Determine the logistic model givenc=12 and the points (0, 9) and (1, 11).


3. Determine the logistic model givenc=7 and the points (0, 2) and (3, 5).
   Answers:


4. The first part involves solving forxwith a known height.

14 = 1 + 420 ·( 0. 9 )x

1 + 4 ·( 0. 9 )x=^2014

( 0. 9 )x=

( 20

14 −^1

)

4

x=log 0. 9

(( 20

14 −^1

)

4

)

=

ln

(( 20

144 −^1 )

)

ln 0. 9 ≈^21.^1995

The second part requires a substitution forx=4.